1. Field of the Invention (Technical Field)
The present invention relates to digital signal processing of spectral data, particularly for measuring gas concentrations from spectral features.
2. Background Art
Note that the following discussion refers to a number of publications by author(s) and year of publication, and that due to recent publication dates certain publications are not to be considered as prior art vis-a-vis the present invention. Discussion of such publications herein is given for more complete background and is not to be construed as an admission that such publications are prior art for patentability determination purposes.
In order to detect or quantify the amount of a substance from a measurement of its spectrum, such as a diode laser spectrum of a single absorption line of a trace gas, several approaches have been used. The simplest is to measure the peak height at the wavenumber of its maximum absorbance, which is related to the gas concentration by Beer""s Law. This method is simple to implement by hand or by computer. However, it makes use of only a small portion of the recorded spectrum and it is sensitive to changes in the baseline signal that may be present in the absence of the analyte. A better method to measure the line amplitude uses the entire recorded spectrum by minimizing the squared deviation between the observed spectrum and a model spectrum. This method is called least squares, and it is widely used in the literature.
When the adjustable parameters in the model spectrum are linear multipliers of some basis functions, then linear least squares analysis can be used. A linear least squares solution can be found in a single step of calculations without the need for iterative algorithms. As a result, linear least squares is fast and robust. The other least squares approach is non-linear least squares. There is no general formula for the solution of a nonlinear least squares problem. Non-linear least squares algorithms require initial guesses for the parameters, which are then adjusted interactively until a desired degree of accuracy is obtained. Non-linear least squares is much slower and less robust than the linear algorithm. However, with non-linear least squares, it is possible to optimize parameters such as the line width and the spectral location of the peak. Such parameters are not linear multipliers of basis functions.
Linear least squares is especially robust and simple to implement when the model does not change from spectrum to spectrum. A constant model arises when the line positions and line widths stay constant. When this condition is met, the most numerically intensive portion of the fit can be computed once; subsequent spectra can be analyzed using this pre-computed matrix, using simpler software on a lower cost processor.
Linear least squares is applied to determining concentrations from spectroscopy by postulating that the observed spectrum can be represented as the weighted sum of one or more spectral features plus background vectors. This postulate implies that both the wavelength or wavenumber position of the spectral feature and its characteristic width are known. However, the wavenumber position can change with time due to variations in spectrometers with temperature or other factors. The line width can change due to variations in the temperature, pressure, or composition of the sample to be measured. In order to accommodate these variations, several approaches have been developed.
D. C. Scott, et al., xe2x80x9cAirborne laser infrared absorption spectrometer (ALIAS-II) for in situ atmospheric measurements of N2O, CH4, CO, HCl, and NO2 from balloon or remotely piloted aircraft platforms,xe2x80x9d Applied Optics 38, 4609-4622 (listed month of publication, July 1999), note that conventional least squares fitting is time consuming, requiring them to adopt a simpler processing strategy of measuring the peak to trough height of the demodulated signals from an airborne diode laser-based instrument. The analysis approach they take to deal with variations in the line width includes prior calculation of a grid of values of the peak to trough height as a function of both temperature and pressure, for unit concentration. However, their approach uses only three points on the spectrumxe2x80x94the maximum and two minima to determine the concentration. It is accurate only when the signal to noise ratio is high and when many digitized steps are sampled on the spectral peak. This in turn requires a fast digitization rate.
A. Fried, et al., xe2x80x9cTunable diode laser ratio measurements of atmospheric constituents by employing dual fitting analysis and jump scanning,xe2x80x9d Applied Optics 32, 821-827 (1993), note the improved precision of linear least squares for determining concentrations from spectra. However, to account for drift of the spectrum within the scan window, several fits are carried out using a variety of center positions, and the fit with the minimum least squares is selected. This method is only accurate to the nearest digital step, so that many digital steps are required.
J. Roths, et al., xe2x80x9cFour-laser airborne infrared spectrometer for atmospheric trace gas measurements,xe2x80x9d Applied Optics 35, 7075-7084 (1996), show that shifts in the center position of a spectral line within a scan can result in increased variance in concentration measurements. They resolve this problem by digitizing the spectrum at closely spaced step intervals and by building a specialized circuit that shifts the peaks to co-align the center positions.
K. S. Booksh, et al., xe2x80x9cMathematical alignment of wavelength-shifted optical spectra for qualitative and quantitative analysis,xe2x80x9d Applied Spectroscopy 50, 139-147 (1996), describe the effects of wavenumber shifts of diode lasers when such lasers are used to excite Raman spectra. They introduce a two step procedure for aligning subsequent spectra, first aligning to the nearest digitized step, then using a Fibbonacci search of possible shifts to interpolate between digitized steps. Although the interpolation is very rapid, it is still an iterative scheme that requires a number of steps to improve the estimate of the line shift.
J. Ivaldi, et al., xe2x80x9cMethod and apparatus for comparing spectra,xe2x80x9d U.S. Pat. No. 5,308,982 (1994), note the effects on analytical precision of spectral shifts in atomic emission and infrared absorption spectroscopies. They disclose a method for least squares fitting of an unknown sample spectrum. In their method, the least squares model includes a vector corresponding to the derivative of the unknown sample spectrum as well as the usual vectors corresponding to the expected peak shape and the background. The derivative vector corrects for the spectral shift. However, this method requires recalculation of the derivative vector after each new measurement of a sample spectrum. As a result, for a continuous monitor, a new model matrix must be constructed and inverted to analyze each new spectrum. The matrix inversion step is usually the most computationally intensive part of the solution. Thus, the procedure disclosed by Ivaldi requires a more expensive, high speed computer for its implementation, or limitations on the update rate.
R. May, xe2x80x9cCorrelation-based technique for automated tunable diode laser scan stabilization,xe2x80x9d Reviews of Scientific Instruments 63, 2922-2926 (1992), describes a method for stabilizing the position of a spectral feature within a scan by using a cross-correlation of the observed spectrum with a reference spectrum that has the peak at the desired reference position, usually the center of the scan. The maximum value of the cross correlation occurs at a position equal to the wavenumber difference between the observed spectrum and the reference spectrum. Feedback is used to adjust the starting point of a subsequent scan to correct any drift of the position that is detected. The cross-correlation algorithm has good signal/noise properties but is relatively slow.
H. Riris, et al., xe2x80x9cDesign of an open path near-infrared diode laser sensor: application to oxygen, water, and carbon dioxide vapor detection,xe2x80x9d Applied Optics 33, 7059-7066 (1994), describe the stabilization of a symmetric spectral feature by measuring the amplitude on the left and right sides of the peak, near the half intensity points. The difference in amplitudes is proportional to spectral shift of the peak; again, feedback is used to stabilize the position. This approach is simple and computationally fast, but has the disadvantage that only a few points are used to determine the spectral shift, resulting in greater susceptibility to noise.
J. McAndrew, et al., xe2x80x9cMethod for stabilizing the wavelength in a laser spectrometer system,xe2x80x9d U.S. Pat. No. 5,742,399 (1998), describe an approach for stabilizing the peak position at its initial value using feedback, except that some other undisclosed algorithm is used instead of the cross-correlation; the computation limited the spectral scan repetition rate to 10 Hz or slower.
T. Guilluk, et al., xe2x80x9cA high-frequency modulated tunable diode laser absorption spectrometer for measurements of CO2, CH4, N2O, and CO in air samples of a few cm3,xe2x80x9d Reviews of Scientific Instruments 68, 230-239 (1997), correct for spectral shifts by adjusting the triggering point of a transient digitizer, so that digitization of the current spectrum is based on the detected peak location in the most recently completed spectrum. This approach requires that only a small part of the scanned spectrum is digitized, thus, much of the scanned region is not available for analysis.
The present invention improves on the art by permitting correction of the concentration measurement for the effects of spectral shift without the need for using non-linear least squares, without abandoning the signal/noise advantages of the least squares technique, and without the need for high speed digitizers, custom circuitry, or high speed computers. It also permits correction of the concentration measurement for errors that can be introduced by changes in the line width of a single line, or by changes in the spectral shape or relative intensities of a band or group of lines as a result of changes in temperature, pressure, or other environmental variables. Furthermore, it provides a means for determining the spectral drift and for stabilizing the wavenumber of a diode laser or other tunable spectrometer element that provides high signal/noise and is computationally efficient.
The present invention is of a spectrographic apparatus and method comprising: providing a spectrometer; digitizing a sample spectrum in a spectral range that includes at least one feature of an analyte; and comparing, via a linear least squares computation, the sample spectrum to a spectrum known to closely approximate the sample spectrum at a reference condition and to one or more derivatives of the known spectrum taken with respect to one or more parameters of the reference condition. In the preferred embodiment, fit coefficients are employed corresponding to terms of the one or more derivatives to correct a concentration determined by the fit coefficient corresponding to the known spectrum. A derivative taken with respect to a center wavenumber of a spectral feature is preferably employed, and the spectral interval of the spectrometer is adjusted employing a weighting factor corresponding to a derivative with respect to wavenumber and negative feedback to stabilize a relative wavenumber position of a spectral peak by adjusting average position of the spectral interval, or the spectral interval of the spectrometer (preferably a diode laser spectrometer) is adjusted employing a weighting factor corresponding to a derivative with respect to wavenumber divided by a weighting factor corresponding to a spectral feature, with the resulting ratio being used with negative feedback to stabilize a relative wavenumber position of a spectral peak by adjusting average position of the spectral interval. Other derivatives usefully employed include derivatives taken with respect to a line width of a spectral feature and derivatives of the sample spectrum taken with respect to analyte temperature and/or pressure. The spectrometer is preferably a diode laser spectrometer, Fourier transform spectrometer, or dispersive spectrometer. The comparing is preferably done by generating a model matrix, multiplying the model matrix by its transpose, inverting the resulting matrix, multiplying this inverted matrix by the transpose of the model matrix, and storing the second resulting matrix (all preferably on a first processor), then using the stored matrix to multiply a sequence of observed spectra to obtain a sequence of concentration measurements (preferably on a second processor, most preferably a digital signal processor or microcontroller). Model spectra are preferably computed at a number of environmental conditions and the known spectrum is preferably the model spectrum whose environmental conditions most closely match those of the analyte.
A primary object of the present invention is to provide: (1) a means to correct the concentration measurement for the effects of spectral shift without the need for using non-linear least squares, without abandoning the signal/noise advantages of the least squares technique, and without the need for high speed digitizers, custom circuitry, or high speed computers; (2) a means for correcting the concentration measurement for errors that can be introduced by changes in the line width of a single line, or by changes in the spectral shape or relative intensities of a band or group of lines as a result of changes in temperature, pressure, or other environmental variables; and (3) a means for determining the spectral drift and for stabilizing the wavenumber of a diode laser or other tunable spectrometer element that provides high signal/noise and is computationally efficient.
A primary advantage of the present invention is that the least squares model and the matrix inverse can be computed before the spectrum is acquired.
A further advantage of the present invention is that accurate results can be obtained with the linear least squares approach even when environmental changes cause small changes in the spectral position or line width of the spectral feature being monitored.